PHYSICS OF PLASMASVOLUME 7, NUMBER 6

JUNE 2000

Shear-flow induced stabilization of kinklike modes
Francesco Volponia)
Graduate School of Engineering, The University of Tokyo, Tokyo 113-8656, Japan

Zensho Yoshidab) and Tomoya Tatsunoc)
Graduate School of Frontier Sciences, The University of Tokyo, Tokyo 113-8656, Japan

共Received 29 November 1999; accepted 14 February 2000兲
The stabilizing effect of a shear flow on a current-driven instability has been studied by means ofKelvin’s representation of spatially-inhomogeneous Galilean transform. Even though conspicuoustransient growth may occur, the mixing effect of the shear flow overcomes the instability and damps
kinklike modes. © 2000 American Institute of Physics. 关S1070-664X共00兲01506-8兴

I. INTRODUCTION

A remarkable finding due to the application of thisscheme was the so-called magnetorotational instability5 in
weakly magnetized accretion disks in astrophysical plasmas.
There are several pioneering theories predicting very transient 共neither exponential nor sinusoidal兲 behavior of plasmas driven by shear flows. In these theories, the free energythat drives the dynamics is primarily due to the shear flow. In
real laboratory systems, as well as in space plasmas, somedifferent free energies can stem from spatial and geometricalinhomogeneities of ambient physical quantities such as magnetic fields. One of the most important neglect committed in
the simple one-dimensional treatment of the ambient fields
共slab model兲 is the curvature of magnetic field lines, whichyields the energy to drive the so-called ‘‘kink instability.’’
Although there are many phenomenological or approximate treatments of the kink modes under the influence of ashear flow, the basic relation between the shear flow and theinstability remains unaddressed on a rigorous basis. The aimof this paper is to develop a theoretical foundation for the
analysis of kinktype instabilities put in a shear flow.
We apply the nonmodal approach for a slab plasma
model. As mentioned above, the naive formulation of a slab
plasma drops the kink-mode driving term. However, invoking a standard technique of incorporating an equivalentmagnetic-field curvature effect, we can introduce an artificial‘‘kink-drive term’’ into the model. Physical applicability ofthe model is rather limited; only spatially localized behavior
is within the scope. The theory, however, captures the essential nature of the competition between the flow shear and theinstability driving effect. We will show that the velocity
shear induces a time dependency on the wave vector, and theresultant ‘‘phase mixing’’ finally overcomes the instability
drive.
In Sec. II we derive the model equations. The nonmodal
method is applied in Sec. III. Results are presented and discussed in Sec. IV.

Shear flows represent a central feature and an open problem in various physical systems. The importance of the roleof shear flows stems from a simple fact; a spatially inhomogeneous flow can drastically alter the modes of their evolution. In other words, depending on the physical characteristics of the system they are embedded in, sheared flows canlead to destabilization or stabilization. In the astrophysicalcontext, for example, differential rotation is considered to be
the priming factor of a dynamo process which might accountfor the presence of strong magnetic fields in planets, stars,
and galaxies. In laboratory plasmas the interest in shear
flows concentrates on their possible stabilizing influence on
unstable modes of fusion plasmas. In spite of their importance, rigorous treatment of the effect of shear has always
been deficient. This is primarily due to the formidable difficulties which plagued the analysis of the differential operators associated with shear flows. These operators indeed turnout to be non-self-adjoint and this fact entails the impossibility of resolving them in terms of orthogonal eigenfunctions.
In recent years, however, the so-called nonmodal approach1,2
is driving important progress in the study of shear flows bothin ordinary hydrodynamics and in magnetohydrodynamics
共MHD兲. The nonmodal method, which was introduced by
Lord Kelvin3 more than a century ago, consists of two basic
methods to solve hyperbolic partial differential equations
共PDEs兲; the characteristics method4 and the Fourier expansion method 共modal method兲. The solution to the characteristic ordinary differential equations 共ODEs兲, associated witha shear flow u0 , gives a new coordinate system 共spatiotemporal兲 on which the convective 共Lagrangian兲 derivative( ⳵ t ⫹u0 •“) reduces into a simple temporal derivative ( ⳵ ␶ ).
In some special cases, the remaining spatial derivatives involved in the PDEs can be converted, by Fourier transform,
into multiplications of some coefficients 共which may be timedependent兲. These two procedures transform a complicated,
spatially inhomogeneous system of PDEs into an easily manageable, temporally inhomogeneous system of ODEs.

II. MODEL OF SHEAR-FLOW MHD
A. Ideal magnetohydrodynamicsa兲Electronic mail: fvolponi@plasma.q.t.u-tokyo.ac.jpb兲Electronic mail: yoshida@plasma.q.t.u-tokyo.ac.jpc兲Electronic mail: tatsuno@plasma.q.t.u-tokyo.ac.jp
1070-664X/2000/7(6)/2314/6/$17.00

Neglecting viscous and resistive effects the dynamics ofa plasma is described by ideal MHD equations,2314© 2000 American Institute of Physics

Phys. Plasmas, Vol. 7, No. 6, June 2000

Shear-flow induced stabilization of kinklike modes␳ 共 ⳵ t ⫹u•ⵜ兲 uÄj⫻B⫺ⵜ p,

共1兲

⳵ t B⫽ⵜ⫻ 共 u⫻B兲 ,共2兲⳵ t ␳ ⫹ⵜ• 共 ␳ u兲 ⫽0,

共3兲

where ␳ and j are, respectively, the mass and current densities, u is the velocity and B is the magnetic field. The convective derivative ( ⳵ t ⫹u•ⵜ) represents the temporal derivative in the frame of the fluid motion. Exploiting the relationj⫽ ␮ ⫺1
0 ⵜ⫻B ( ␮ 0 is the vacuum permeability兲 and assuming
a barotropic relation ⵜ p⫽c s2 ⵜ ␳ (c s is the sound speed兲 wecan eliminate j and p from Eq. 共1兲.
We introduce the following set of dimensionless variables:x⫽lx̂,␳ ⫽ ␳ 0 ␳ˆ ,B⫽B T B̂,t⫽ 共 l/c A 兲 t̂ ,u⫽c A û,

共4兲where l is a characteristic length scale of the system, B T and␳ 0 are the representing values of the magnetic field and density, respectively, and c A ⫽B T /( ␮ 0 ␳ 0 ) 1/2 is the Alfvén speed.
The dimensionless form of 共1兲–共3兲 is␳ˆ 共 ⳵ t̂ ⫹û•ⵜ 兲 û⫽ 共 ⵜ⫻B̂兲 ⫻B̂⫺ ␤ ⵜ ␳ˆ ,共5兲⳵ t̂ B̂⫽ⵜ⫻ 共 û⫻B̂兲 ,共6兲⳵ t̂ ␳ˆ ⫹ⵜ• 共 ␳ˆ û兲 ⫽0,

共7兲

where ␤ ⫽(c s /c A ) . In what follows we will drop the hat inorder to simplify the notation.
We decompose the physical quantities into their equilibrium and small perturbative components to write2

BÄB0 ⫹B̃,

uÄu0 ⫹ũ,␳ ⫽ ␳ 0 ⫹˜␳ ,共8兲

where the subscript ‘‘0’’ denotes equilibrium fields, and thetilde, the perturbations fields. Linearizing 共5兲–共7兲, we obtain␳ 0 共 ⳵ t ⫹u0 •ⵜ 兲 ũ⫹˜␳ u0 •ⵜu0 ⫹ ␳ 0 ũ•ⵜu0⫽ 关共 ⵜ⫻B̃兲 ⫻B0 ⫹ 共 ⵜ⫻B0 兲 ⫻B̃兴 ⫺ ␤ ⵜ˜␳ ,共9兲⳵ t B̃⫽ⵜ⫻ 共 ũ⫻B0 兲 ⫹ⵜ⫻ 共 u0 ⫻B̃兲 ,共10兲⳵ t˜␳ ⫽⫺ⵜ• 共 ␳ 0 ũ⫹˜␳ u0 兲 .

共11兲Here, we remark the importance of the inhomogeneous ambient flow u0 . The standard linear MHD theory assumes thatu0 ⫽constant (⯝0). Then, the generator of the dynamics becomes self-adjoint, and this fact allows us to apply the modal
method 共spectral analysis兲. When u0 is not constant, however, the system becomes non-self-adjoint, and this introduces profound difficulties in the spectral analysis of theideal MHD operator.6,7
B. One-dimensional slab model

We consider a one-dimensional slab plasma. We express
all quantities in Cartesian x – y – z coordinates where, to compare with tokamak geometry, x parallels the radial, y thepoloidal, and z the toroidal coordinates. The x is the directionin which the ambient fields vary. The perturbation are functions of x and y, while they are homogeneous with respect
to z.

2315

We assume that the equilibrium density is a positive
constant and set ␳ 0 ⫽1. To model a tokamak plasma, we
assume 兩 B 0 y /B 0 z 兩 ⫽ ⑀ Ⰶ1, and set B T 共representative value ofB) ⫽B 0 z . The equilibrium magnetic field is given by

冉冊
冉 冊
0

B0 ⫽ ⫺ ⑀ .
1

共12兲

The shear flow u0 is an incompressible flow given by0

u0 ⫽ ⫺Ax ,0

共13兲

where A represents the strength of the velocity shear.
By assuming a symmetry ( ⳵ z ⫽0) and incompressibility(ⵜ"ũ⫽0), we may writeB̃⫽ⵜ ␺ ⫻ⵜz⫹B̃ z ⵜz,ũ⫽ⵜ ␾ ⫻ⵜz⫹ũ z ⵜz.

共14兲A strong toroidal field (B 0 z ) allows us to assume B̃ z ⫽ũ z⫽0 共tokamak ordering兲. Then the perturbations are represented by two fields; ␺ (z component of the vector potential兲and ␾ 共velocity streamfunction兲 that giveB̃⫽ⵜ ␺ ⫻ⵜz,ũ⫽ⵜ ␾ ⫻ⵜz.

共15兲

Under this assumption, the perturbations 共15兲 obey a selfconsistent system of linear MHD equations, which describes
the shear Alfvénic mode. We note that this mode of perturbations is decomposed from other compressional modes
共slow and fast modes兲 and forms a closed system. A kinkinstability, that is an unstable shear Alfvénic mode, becomesmost unstable when it is decoupled from compressionalmodes that have positive energies.
Let us introduce the ‘‘kink-drive term’’ into the model.
As mentioned above, we have to assume an artificial ‘‘curvature’’ of the magnetic field lines, which yields the energy
to drive the kink mode.
In the evolution equation 共9兲, this effect appears as theterm (ⵜ⫻B0 )⫻B̃. Although ⵜ⫻B0 ⫽0 for the present B0
关see 共12兲兴, we may introduce an artificial curvature by calculating the curl derivative in the cylindrical coordinates.
For a general vector field A, we have1
共 ⵜ⫻A兲 r ⫽ ⳵ ␪ A z ⫺ ⳵ z A ␪ ,r
共 ⵜ⫻A兲 ␪ ⫽ ⳵ z A r ⫺ ⳵ r A z ,1
共 ⵜ⫻A兲 z ⫽ 共 ⳵ r 共 rA ␪ 兲 ⫺ ⳵ ␪ A r 兲 .
rHere x parallels the radial coordinate r, while y and z correspond to the poloidal ( ␪ ) and toroidal coordinates. Using
these correspondences, we obtain
共 ⵜ⫻B0 兲 x ⫽0,
共 ⵜ⫻B0 兲 y ⫽0,2316

Phys. Plasmas, Vol. 7, No. 6, June 2000

Volponi, Yoshida, and Tatsuno

The system 共26兲–共27兲 is homogeneous with respect tothe coordinates ␰ and ␩ . Fourier transform of ␾ and ␺ , thus,
yields ‘‘good quantum numbers’’ k ␰ and k ␩ . Let us write1
1
共 ⵜ⫻B0 兲 z ⫽⫺ ⳵ x 共 x 兲 ⫽⫺ .
x
x
The expression for (ⵜ⫻B0 )⫻B̃ becomes␺ 共 ␰ , ␩ , ␶ 兲 ⫽ ␺ˆ 共 ␶ 兲 e i(k ␰ ␰ ⫹k ␩ ␩ ) ,1
共 ⵜ⫻B0 兲 ⫻B̃⫽⫺ ⵜ ␺ .
x

共28兲

After taking the curl of the momentum equation 共9兲 and thecurl⫺1 of Faraday Law 共10兲 in Cartesian coordinates we obtain⳵ t 共 ⌬ ␾ 兲 ⫺Ax ⳵ y 共 ⌬ ␾ 兲 ⫽⫺ ⑀ ⳵ y 共 ⌬ ␺ 兲 ⫺ ⑀ x ⫺2 ⳵ y ␺ ,共16兲⳵ t ␺ ⫺Ax ⳵ y ␺ ⫽⫺ ⑀ ⳵ y ␾ ,共17兲where ⌬⫽ ⳵ 2x ⫹ ⳵ 2y is the Laplacian. The second term on theright-hand side of 共16兲 brings about the kink 共current-driven兲
instability.

We introduce a spatio-temporal coordinate transform8
from the fixed 共Eulerian兲 reference frame x – y – z to the local
moving frame 共Lagrangian兲 going with the mean shear flow,

␰ ⫽x,

共18兲

␩ ⫽y⫹Axt,

共19兲

␶ ⫽t.

共20兲

The above transformation represents the characteristic raysof the convective derivative operator D/Dt⬅ ⳵ t ⫹u0 •ⵜ. Indeed it can be readily obtained from the characteristic ODE
associated with D/Dt. Solving冉冊 冉 冊Then, the system of PDEs 共26兲 and 共27兲 reduces into a system of ODEs,dˆ兲
共关共 k ␰ ⫹k ␩ A ␶ 兲 2 ⫹k ␩2 兴 ␾d␶ˆ⫽⫺ ⑀ ik ␩ 共关共 k ␰ ⫹k ␩ A ␶ 兲 2 ⫹k ␩2 兴 ⫺R ⫺2
0 兲␺,

共21兲

冉冊 冉冊
y

共 0 兲⫽

␰k x ⫽k ␰ ⫹Ak ␩ ␶ ,

␩

共22兲

yields 共18兲 and 共19兲.
Equations 共18兲, 共19兲, 共20兲 induce the following transformation of the partials:⳵ x ⫽ ⳵ ␰ ⫹At ⳵ ␩ ,共23兲⳵ y⫽⳵␩ ,共24兲

⳵ t ⫽ ⳵ ␶ ⫹Ax ⳵ ␩ .

共25兲

The convective derivative D/Dt⫽ ⳵ t ⫺Ax ⳵ y reduces into a
simple ‘‘temporal’’ derivative ⳵ ␶ .
Equations 共16兲 and 共17兲 now read⳵ ␶ 共 ⌬ ⬘ ␾ 兲 ⫽⫺ ⑀ 关 ⳵ ␩ 共 ⌬ ⬘ ␺ 兲 ⫺R ⫺2
0 ⳵ ␩␺ 兴 ,共26兲

⳵ ␶ ␺ ⫽⫺ ⑀ ⳵ ␩ ␾ ,共27兲

where ⌬ ⬘ ⫽ ⳵ ␰2 ⫹(1⫹A 2 t 2 ) ⳵ ␩2 ⫹2At ⳵ ␰ ⳵ ␩ is the Laplacian
represented in the new coordinate system and R 0 is a constant number representing the curvature radius of the system.
We note that the present model of field-line curvature assumes radially localized perturbations.

k y ⫽k ␩ ,共31兲

where k x and k y are, respectively, the wave numbers in the x
and y coordinates. Equations 共31兲 clearly shows that the k xvaries with time, implying that the shear flow deforms themode of perturbation.
After simple manipulations, we obtain the following second order ODE for ␺ˆ ,
d2
dT

2

␺ˆ ⫹2a冋k x 共 T 兲 /k yd␺ˆ
共 k x 共 T 兲 /k y 兲 ⫹1 dT
2⫹ ⑀ 1⫺
2,

共30兲

We point out that Eqs. 共23兲, 共24兲, 共25兲, and 共28兲 imply thefollowing transformation for the wave numbers:

subject to an initial condition
x

共29兲

d
␺ˆ ⫽⫺ ⑀ ik ␩ ␾ˆ .
d␶III. NONMODAL REPRESENTATION

0
d x
⫽
,
dt y
⫺Ax

␾ 共 ␰ , ␩ , ␶ 兲 ⫽ ␾ˆ 共 ␶ 兲 e i(k ␰ ␰ ⫹k ␩ ␩ ) .

r ⫺20
共 k x 共 T 兲 /k y 兲 2 ⫹1册␺ˆ ⫽0,

共32兲

where k x (T)/k y ⫽ ␬ ␰ ⫹aT and T,a,r 0 , ␬ ␰ are the normalized
quantities defined byT⫽ ␶ k ␩ ,a⫽A 共 k ␩ 兲 ⫺1 ,r 0 ⫽R 0 k ␩ ,␬ ␰ ⫽k ␰ k ␩⫺1 .

共33兲

Defining

␮ 共 T 兲 ⫽2a

k x 共 T 兲 /k y
共 k x 共 T 兲 /k y 兲 ⫹12

冋

⍀ 2 共 T 兲 ⫽ ⑀ 2 1⫺

⫽2a

r ⫺2
0
共 k x 共 T 兲 /k y 兲 2 ⫹1

␬ ␰ ⫹aT
共 ␬ ␰ ⫹aT 兲 2 ⫹1

册 冋

⫽ ⑀ 2 1⫺

we write 共32兲 as
d2
dT

2

␺ˆ ⫹ ␮ 共 T 兲d␺ˆ ⫹⍀ 2 共 T 兲 ␺ˆ ⫽0.
dT

共34兲

,
r ⫺2
0

册

,
共 ␬ ␰ ⫹aT 兲 2 ⫹1
共35兲

Phys. Plasmas, Vol. 7, No. 6, June 2000

Shear-flow induced stabilization of kinklike modes

2317

TABLE I. Relation between ␬ ␰ and ␮ (T).
Case

␬␰

␮ (T) at small T

␮ (T) at large T

A
B

⫹
⫺

⫹
⫺

⫹
⫹

IV. ANALYSIS OF THE EVOLUTION OF THE
PERTURBATIONS

We pursue the analysis of Eq. 共32兲 in three steps. Firstwe will examine the effect of the kinklike driving term andthe shear term separately and then we will investigate their
combined action.
A. No shear flow „Alfvén wave and kink instability…

Assuming a⫽0 共zero flow兲, Eq. 共32兲 becomes
d2
dT

冉

␺ˆ ⫹ ⑀ 2 1⫺
2

r ⫺2
0

␬ ␰2 ⫹1冊

␺ˆ ⫽0.

共36兲

The curvature radius r 0 is the characteristic parameter that
dominates the behavior of the solution. For sufficiently larger 0 共small curvature兲, i.e., r 20 ⬎1/( ␬ 2␰ ⫹1), 共36兲 describes anharmonic oscillator, which represents the Alfvén wave. For asmall r 0 共large curvature兲, we may have an exponentiallygrowing solution, representing the kink instability. The transition from a stable oscillation 共Alfvén wave兲 to an unstablemotion 共kink instability兲 depends on the wave number ␬ ␰
共⫽const. if a⫽0); r 20 ⫽1/( ␬ 2␰ ⫹1) is the critical number.
B. No kink drive „shear-flow damping…

In the absence of the kink-driving term (r ⫺20 ⫽0), Eq.
共32兲 reduces into
d2
dT

2

␺ˆ ⫹2a␬ ␰ ⫹aTd␺ˆ ⫹ ⑀ 2 ␺ˆ ⫽0.
共 ␬ ␰ ⫹aT 兲 ⫹1 dT
2FIG. 2. Evolution of ␺ˆ for r ⫺20 ⫽0, ␬ ␰ ⫽⫺4, ⑀ ⫽0.1, and a⫽0.5. As initialvalues we have chosen ␺ˆ (0)⫽0.04 and (d/dT) ␺ˆ (0)⫽0.05.

This behavior of the solution, which is due to the phasemixing effect induced by the shear flow, presents an analogyof the Landau damping that is induced by the shear flow inthe coordinate-velocity phase space of the kinetic theory. On
the contrary, a negative value of ␮ (T) yields amplification of␺ˆ .
The frictional coefficient ␮ (T) depends on the parameters ␬ ␰ and a. In the following analysis, without loss ofgenerality, we will take the signs of A and k ␩ to be positive
共then a is positive兲. In Table I we summarize the dependenceof ␮ (T) on the sign of ␬ ␰ . We observe that, in any case,␮ (T) is positive for large T. This implies that the solution of
共37兲 is finally damped. In Case A, ␺ˆ shows only damping
oscillations 共Fig. 1兲, however, in Case B, a transient amplification of ␺ˆ occurs 共Fig. 2兲. The maximum of the amplitude
is reached around the time when ␮ (T) changes its sign.

共37兲

The behavior of the solution depends on the sign of the coefficient ␮ (T) defined in 共34兲. Invoking the analogy of 共37兲with Newton’s equation of motion, we can interpret the second term on the right-hand side of 共37兲 as a frictional force.
A positive ␮ (T) damps the oscillation of ␺ˆ 共Alfvén wave兲.

FIG. 1. Evolution of ␺ˆ for r ⫺20 ⫽0, ␬ ␰ ⫽4, ⑀ ⫽0.1, and a⫽0.5. As initialvalues we have chosen ␺ˆ (0)⫽0.04 and (d/dT) ␺ˆ (0)⫽0.05.

FIG. 3. Evolution of k for positive and negative values of k ␰ .

2318

Phys. Plasmas, Vol. 7, No. 6, June 2000

Volponi, Yoshida, and Tatsuno

FIG. 4. Evolution of ␺ˆ for r ⫺2
0 ⫽400, ␬ ␰ ⫽4, ⑀ ⫽0.1, and a⫽0.5. As initialvalues we have chosen ␺ˆ (0)⫽0.04 and (d/dT) ␺ˆ (0)⫽0.05.

FIG. 6. Evolution of ␺ˆ for r ⫺2
0 ⫽400, ␬ ␰ ⫽⫺4, ⑀ ⫽0.1, and a⫽5. As initialvalues we have chosen ␺ˆ (0)⫽0.04 and (d/dT) ␺ˆ (0)⫽0.05.

Let us see the evolution of the wave vector k 共in the x – ycoordinates兲 and its relation to the behavior of ␺ˆ . The initialconfiguration is k(0)⫽(k ␰ ,k ␩ ). First we consider the casewhen k ␰ is positive. By 共31兲 we find that k x increases monotonically with time, i.e., k is stretched in the positive k x
direction 共Case A of Fig. 3兲. This case corresponds to thesimple damping oscillations. When k ␰ is negative, the evolution of k experiences two distinct phases. For 0⭐ ␶ ⭐ ␶ *⫽⫺ ␬ ␰ /a, k x shrinks until it becomes zero, and during thisphase, the amplification of the perturbation proceeds. For ␶⭓ ␶ * , the absolute value of k x grows and k is stretched in thepositive k x direction 共Case B of Fig. 3兲. This stretch of k x
yield the phase mixing damping.

共proportional to a) diminishes the magnitude of the kinkdriving term 共for large T, it becomes proportional to T ⫺2 ).
On the other hand, ␮ (T) becomes positive for large T.
Therefore, the phase mixing effect induced by the shear flow
is asymptotically stronger than the kink mode destabilization.
Let us study the behavior of the solution of 共32兲 in moredetail. Consider a small value of r 0 共high r ⫺20 ), so that the
mixing damping is negligible. The solution grows until⍀ 2 (T) becomes positive. Writing ⍀ 2 (T̄)⫽0, this criticaltime T̄ is given by the positive root ofT̄⫽C. Competition between the kink drive and the shearflow induced damping

The discussion in the previous subsections shows thefollowing relations:⍀ 2 共 T 兲 ⬎0: oscillatory behavior,⍀ 2 共 T 兲 ⬍0: instability,

␮ 共 T 兲 ⬎0: damping,␮ 共 T 兲 ⬍0:

amplification.

We have seen that the kink-driving term 共proportional to)contributes negatively in ⍀ 2 (T). The shear-flow effectr ⫺20

FIG. 5. Evolution of ␺ˆ for r ⫺2
0 ⫽400, ␬ ␰ ⫽⫺4, ⑀ ⫽0.1, and a⫽0.5. As
initial values we have chosen ␺ˆ (0)⫽0.04 and (d/dT) ␺ˆ (0)⫽0.05.

⫺ ␬ ␰ ⫾ 冑r ⫺2
0 ⫺1
a

.

共38兲

We can then single out in the time domain an instability
region 关 0,T̄ 兴 of growth for the oscillations.1 As shown in
Figs. 4 and 5, the growth can be very rapid, especially for
negative k ␰ . Equation 共38兲 suggests that a strong shear candiminish the time interval of the growth, resulting in a reduction of the maximum amplitude. This phenomenon is shownin Figs. 6 and 7 where we compare two different shears. The
strong stabilizing effect of the shear is clearly visible. In
Figs. 8 and 9 we consider two different configurations of themagnetic field. The results suggest that for small values of ⑀

FIG. 7. Evolution of ␺ˆ for r ⫺2
0 ⫽400, ␬ ␰ ⫽⫺4, ⑀ ⫽0.1, and a⫽50. As
initial values we have chosen ␺ˆ (0)⫽0.04 and (d/dT) ␺ˆ (0)⫽0.05.

Phys. Plasmas, Vol. 7, No. 6, June 2000

Shear-flow induced stabilization of kinklike modes

FIG. 8. Evolution of ␺ˆ for r ⫺2
0 ⫽400, ␬ ␰ ⫽⫺4, ⑀ ⫽0.01, and a⫽0.5. As
initial values we have chosen ␺ˆ (0)⫽0.04 and (d/dT) ␺ˆ (0)⫽0.05.

共i.e., B0 and u0 close to a condition of orthogonality兲 thekinklike modes are more effectively stabilized.
V. SUMMARY

A shear flow brings about the strong stabilizing effectthat can overcome the kink-type instability at a sufficiently
large time. The analysis of this process requires a nonmodal
method. This stabilizing effect is due to the deformation
共stretching兲 of the mode of the instability. To highlight thispoint, let us compare the above-mentioned results with thecalculations for a rigid 共nonsheared兲 flow. For a rigid flow
(0,⫺A,0) the transformation to a Lagrangian system is given
by a Galilean transform,␰ ⫽x,␩ ⫽y⫹At,

␶ ⫽t,

共39兲

⳵ y⫽⳵␩ ,⳵ t ⫽ ⳵ ␶ ⫹A ⳵ ␩ .

FIG. 10. Spatial dependence of the real part of the perturbation ␺ relative to␺ˆ in Fig. 6 at four different times.

Thus, for a rigid flow, the coordinate transform does not
induce any time dependence of k. In Fig. 10, we show thespatial profile of the real part of ␺ 共relative to the simulationin Fig. 6兲 at four different times. We observe that the mode isstrongly deformed by the shear flow.
The evolution of a non-self-adjoint system is generally
very complicated, and can be very different from usual oscillatory or exponential behavior. Transient amplification of
perturbations may occur in an essentially stable system.9 On
the contrary, the mixing effect may induce a strong damping
effect that dominates the long-term behavior of the system.
Even though conspicuous transient growth for the perturbation fields are observed, an increase in the magnitude of theshear has a huge impact on the reduction of their maximum
amplitude.
ACKNOWLEDGMENTS

which transforms the partial derivatives as⳵ x⫽ ⳵ ␰ ,

2319

共40兲

The authors are grateful to Professor Swadesh M. Mahajan and Dr. Andria Rogava for their discussions and suggestions. This work was supported by Grant-in-Aid for Scientific Research from the Japanese Ministry of Education,
Science, and Culture No. 09308011.
1

FIG. 9. Evolution of ␺ˆ for r ⫺2
0 ⫽400, ␬ ␰ ⫽⫺4, ⑀ ⫽0.3, and a⫽0.5. As
initial values we have chosen ␺ˆ (0)⫽0.04 and (d/dT) ␺ˆ (0)⫽0.05.

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